Least-square approach for singular value decompositions of scattering problems

TL;DR

This paper introduces a general least-square method for applying singular value decomposition to scattering problems, enabling efficient low-rank approximations that accurately capture scattering observables across different frameworks.

Contribution

It presents a novel least-square approach for SVD in scattering problems, facilitating low-rank reductions in various few- and many-body frameworks.

Findings

Successfully solved the Lippmann-Schwinger equation in factorized form.

Low-rank approximations accurately reproduce scattering observables.

Method applicable to other tensor factorization frameworks.

Abstract

It was recently observed that chiral two-body interactions can be efficiently represented using matrix factorization techniques such as the singular value decomposition. However, the exploitation of these low-rank structures in a few- or many-body framework is nontrivial and requires reformulations that explicitly utilize the decomposition format. In this work, we present a general least-square approach that is applicable to different few- and many-body frameworks and allows for an efficient reduction to a low number of singular values in the least-square iteration. We verify the feasibility of the least-square approach by solving the Lippmann-Schwinger equation in factorized form. The resulting low-rank approximations of the $T$ matrix are found to fully capture scattering observables. Potential applications of the least-square approach to other frameworks with the goal of employing tensor factorization techniques are discussed.